Appendix A

Determination of the Size and Mass of the Universe.


To start this derivation, consider the rate of expansion of the Universe in Phase II of its evolution, exemplified by the recession velocity of a random galaxy, a distance of si from the centre, as given by [2], Eq.(3.22)
×
s
 
i 
 = ui æ
è 		gmu
su3
ö
ø 		1/2

 
 si
(A.1)
In [2] this was simplified to
×
s
 
i 
 = H0 si
(A.2)

where H0 is the value of Hubble's "constant" at si, and in which ui had been approximated by unity. In (A.2) the latter approximation can be avoided by using the current empirical value for H0, which automatically includes the correct effect for ui. However, this value applies only at the cosmologically applicable time of tC as fully explained in [2], Section 3.5.

Now if (A.2) is to apply to the boundary of the Universe then it becomes

×
s
 
u 
 = H0 su
(A.3)

and in using the above value of H0 in (A.3), the spatial gradient effect between si and su, both at tC, has been ignored. This is effectively once again assuming ui=1 over this distance. Thus (A.3) includes the same degree of approximation as was used in [2] to determine the theoretical value of H0. Therefore, treating H0 in (A3) as a constant, (A.3) can be integrated to give

Ln( su ) = H0 tC + k
(A.4)
where Ln represents Naperian logarithms.

In (A.4) the cosmologically applicable time tC is the time the Universe has been expanding from the point of inflexion, and the constant of integration is therefore the Naperian logarithm of the radius of the point of inflexion from the centre. In a next paper it will be shown that this radius is equal to twice the gravitational radius. Thus in (A.4) when tC = 0

Ln( 2au ) = k
(A.5)
which when substituted into (A.4) gives
su = 2au exp( H0 tC )
(A.6)

Eq.(A.6) is a simplified relationship between the radius of the Universe and the time at tC in Phase II of its evolution from the point of inflexion. To eliminate au in (A.6) substitute

au = gmu
c2
 = 4pgru su3
3c2
(A.7)
Eq.(A.6) then becomes
su = æ
è 		3c2
8pgru exp( H0 tC )
 ö
ø 		1/2

 
(A.8)

At the time tC in (A.8), H0 is known to be 22.6 Km/sec/106     L.Y. for which the corresponding value of ru from [2] Eq.(3.24) is 2.1E-29 gm/cm3 and all other terms are constants. Thus if tC were known, a value of su at tC could be calculated. An estimate of tC is constructed as follows.

In surveying the recession velocities of the distant galaxies and the density of the Universe, if the range of observations was between say 4 and 8 billion L.Y., and assuming a uniform distribution of observations between that range, then the value of tC would be at the median point of the range and can accordingly be estimated to be approximately 6 billion years ago. In [2] it was stated that the evolution of the Universe was probably some 15 billion years into Phase II from the point of inflexion. The cosmologically applicable time in (A.8), tC, is then estimated as the difference between the above two numbers, i.e. 9 billion years.

Inserting this value, and those for H0 and ru above, together with the known values for c and g as previously stated, into (A.8) then yields

su = 5.7E27 cm @ 6.03 L.Y.
(A.9)

as the radius of the Universe at tC. The mass of the Universe can now be calculated using the standard formula to be

mu = 4
3
 pru su3 = 1.65E55 gm.
(A.10)
The values in (A.9) and (A.10) can now be inserted into Table 2.

Finally, as a matter of interest, au the gravitational radius of the Universe can now be established as

au = gmu
c2
 = 1.22E27 cm = 1.29E9 L.Y.
(A.11)

It should be noted that all the numbers generated here, are very approximate estimates resulting from approximations in the expressions used and the uncertainties in such parameters as ru, H0 and tC. However, these numbers are so large that errors of even one or two magnitudes are insignificant when used in Table 2 and Fig.1 for the purpose intended.


C2 Version 1.2.1
Ó P.G.Bass, December 2009
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